Densitometers, colorimeters, spectrophotometers, radiometers and chroma meters all perform the same type of physical measurement. They each calculate one or more weighted integrations of optical energy over wavelength. In the case of densitometers, colorimeters, some radiometers and chroma meters, a small number of weighted integrations are normally performed. The weighting function for densitometers and colorimeters is usually the product of the spectral intensity of a light source, a filter, the sensitivity profile of a detector, and the reflection or transmission of a sample.
For chroma meters and radiometers the sample is a radiant source, and the weighting function is the product of the spectral intensity of the source, a filter parameter, and the spectral sensitivity of a detector. In the current practice for measurement of reflectance or transmission, the light source is a broad band emitter such as a tungsten lamp, and the detector has wide range of sensitivity. In general the same light source and detector profiles are used for each weighted integration and only the filter profile changes. These instruments are usually calibrated in use by using a sample of known transmittance or reflectance. Spectrophotometers and spectral radiometers are generally capable of reporting tens to thousands of weighted integrations For spectrophotometers and spectral radiometers, the weighting functions usually each have a common shape which is ideally a narrow triangle. Multiple weighted integrations are obtained by choosing multiple positions for the center of the weighting function along the wavelength axis. Spectrophotometers generally use broad band light sources and detectors, but because they may span a greater wavelength range, they may have multiple sources or detectors. The weighting function is provided by continuous filters, by multiple filters, by monochromators, by spectrographs, or by interference techniques. Densitometers, colorimeters and chroma meters are designed to produce specific weighed integrations. For densitometers used in graphic arts, the weighting of the integrations is intended to optimize the response to the reflectance of standard inks used in printing. For colorimeters and chroma meters the weighting of the integrations is intended to generate chromaticity coordinates as defined by CIE or ASTM. Spectrophotometers and spectral radiometers are intended to allow the reflectance, transmittance or intensity spectrum with reference to wavelength of the sample to be represented as a set of points, or a curve, or a spectrum. As all of these weighted integrations are linear sums over wavelength, each integration may be considered to be a vector in a common vector space. A particular set of weighted integrations will correspond to a collection of vectors from this space which form a subspace. Such subspaces are said to be spanned by the set of weighted integrations from which their component vectors may be calculated. In these terms each type of instrument can be said to measure the subset of the space which is spanned by the vectors represented by its weighted integrations. In general, the weighted integrations designed into densitometers, colorimeters, and chroma meters span a subset of the space which is spanned by the weighting functions designed into spectrophotometers and spectral radiometers. When colorimeters, densitometers, and chroma meters are designed to published standards, the specified integrations are stated in terms of sums over spectra, and these instruments are usually calibrated against the measurements of a high grade spectrophotometer or spectral radiometer or against samples for which the relectance, transmission, or spectral intensity as a function of wavelength has been independently determined. Given that each set of these weighted integrations can be expressed as subsets of the same vector space it follows that any set of weighted integrations can be transformed into any other set within the subset of the space spanned by both sets, and further it follows that this calculation will be a linear transformation which may be performed as a matrix multiplication. This is what occurs when chromaticity coordinates or densities are calculated from spectral data produced by a spectrophotometer or spectral radiometer. It also follows that there will be a collection of other sets of weighted integrations which will span the subset of the space spanned by chromaticity coordinates and by density specifications. Any such subset may be measured and used to calculate chromaticity coordinates or density functions.